TOWARDS A PHILOSOPHY OF REAL MATHEMATICS In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathe- matics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. His highly original book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines, and points clearly to the ways in which this can be done. david corfield holds a Junior Lectureship in Philosophy of Science at the University of Oxford. He is co-editor (with Jon Williamson) of Foundations of Bayesianism (2001), and he has pub- lished articles in journals including Studies in History and Philosophy of Science and Philosophia Mathematica. TOWARDS A PHILOSOPHY OF REAL MATHEMATICS DAVID CORFIELD The Pitt Building, Trumpington Street, Cambridge, United Kingdom The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org © David Corfield 2004 First published in printed format 2003 ISBN 0-511-04270-1 eBook (netLibrary) ISBN 0-521-81722-6 hardback From the east to western Ind, No jewel is like Rosalind. Contents Prefacepage ix 1 Introduction: a role for histor y 1 part i human and artificial mathematicians 2 Communicating with automated theorem provers 37 3 Automated conjecture formation 57 4 The role of analogy in mathematics 80 part ii plausibility, uncertainty and probability 5 Bayesianism in mathematics 103 6 Uncer tainty in mathematics and science 130 part iii the growth of mathematics 7 Lakatos’s philosophy of mathematics 151 8 Beyond the methodology of mathematical research programmes 175 9 The impor tance of mathematical conceptualisation 204 part iv the interpretation of mathematics 10 Higher-dimensional algebra 237 Appendix 271 Bibliography 274 Index 286 vii Preface I should probably not have felt the desire to move into the philosophy of mathematics had it not been for my encounter with two philosophical works. The first of these was Imre Lakatos’s Proofs and Refutations (1976), a copy of which was thrust into my hands by a good friend Darian Leader, who happens to be the godson of Lakatos. The second was an article entitled ‘The Uses and Abuses of the History of Topos Theory’ by Colin McLarty (1990), a philosopher then unknown to me. What these works share is the simple idea that what mathematicians think and do should be important for philosophy, and both express a certain annoyance that anyone could think otherwise. Finding a post today as a philosopher of mathematics is no easy task. Finding a post as a philosopher of mathematics promoting change is even harder. When a discipline is in decline, conservatism usually sets in. I am, therefore, grateful beyond words to my PhD supervisor, Donald Gillies, both for his support over the last decade and for going to the enormous trouble of applying for the funding of two research projects, succeeding in both, and offering one to me. The remit of the project led me in directions I would not myself have chosen to go, especially the work reported in chapters 2 and 3, and I rather think chapters 5 and 6 as well, but this is often no bad thing. I am thus indebted to the Leverhulme Trust for their generous financial support. Thanks also to Jon Williamson, the other fortunate recipient, for discussions over tapas. Colin McLarty has provided immense intellectual and moral support over the years, and also arranged a National Endowment of the Humanities Summer Seminar where sixteen of us were allowed the luxury of talking philosophy of mathematics for six weeks in the pleasant surroundings of Case Western Reserve University. My thanks to the NEH and to the other participants for making it such an enjoyable experience. I should also like to acknowledge the helpful advice of Ronnie Brown, Jeremy Butterfield, James Cussens, Matthew Donald, Jeremy Gray, Colin ix x Preface Howson, Mary Leng, Penelope Maddy, Stephen Muggleton, Madeline Muntersbjorn, Jamie Tappenden, Robert Thomas and Ed Wallace. This book could only have benefited from greater exposure to the intellec- tual ambience of the History and Philosophy of Science Department in Cambridge, where the writing was finished. Unfortunately time was not on my side. I only hope a little of the spirit of the department has trickled through into its pages. Hilary Gaskin at Cambridge University Press has smoothed the path to publication. Four of the chapters are based on material published else- where. Chapter 5 is based on my chapter in Corfield and Williamson 2001, Foundations of Bayesianism, Kluwer. Chapters 7 and 9 are based on papers of the same title in Studies in the History and Philosophy of Science, 28(1): 99–121 and 32(3): 507–33. Chapter 8 is likewise based on my article in Philosophia Mathematica 6: 272–301. I am grateful to Kluwer, Elsevier and Robert Thomas for permission to publish them. I should like to thank J. Scott Carter and Masahico Saito for kindly providing me with the figure displayed on the cover. It shows one of the ingenious ways they have devised of representing knotted surfaces in four- dimensional space. In chapter 10 we shall see how this type of representation permits diagrammatic calculations to be performed in higher-dimensional algebra. Love and thanks to Oliver, Kezia and Diggory for adding three more dimensions to my life beyond the computer screen, and to my parents for all their support. This book I dedicate to Ros for fourteen years of sheer bliss. The publisher has used its best endeavours to ensure that the URLs for external websites referred to in this book are correct and active at the time of going to press. However, the publisher has no responsibility for the web- sites and can make no guarantee that a site will remain live or that the con- tent is or will remain appropriate. chapter 1 Introduction: a role for history To speak informatively about bakery you have got to have put your hands in the dough. (Diderot, Oeuvres Politiques) The history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty. (Lakatos, Proofs and Refutations) 1.1 real mathematics To allay any concerns for my mental health which the reader may be feeling if they have come to understand from the book’s title that I believe math- ematics based on the real numbers deserves singling out for philosophical treatment, let me reassure them that I mean no such thing. Indeed, the glorious construction of complex analysis in the nineteenth century is a paradigmatic example of what ‘real mathematics’ refers to. The quickest way to approach what I do intend by such a title is to explain how I happened upon it. Several years ago I had been invited to talk to a philosophy of physics group in Cambridge and was looking for a striking title for my paper where I was arguing that philosophers of mathematics should pay much closer attention to the way mathematicians do their research. Earlier, as an impecunious doctoral student, I had been employed by a tutorial college to teach eighteen-year-olds the art of jumping through the hoops of the mathematics ‘A’ level examination. After the latest changes to the course ordained by our examining board, which included the removal of all traces of the complex numbers, my colleagues and I were bemoaning the reduction in the breadth and depth of worthwhile content on the syllabus. We started playing with the idea that we needed a campaign for the teaching of real mathematics. For the non-British and those with no interest in beer, the allusion here is to the Campaign for Real Ale (CAMRA), a movement dedicated to maintaining traditional brewing 1 2 Towards a Philosophy of Real Mathematics techniques in the face of inundation by tasteless, fizzy beers marketed by powerful industrial-scale breweries. From there it was but a small step to the idea that what I wanted was a Campaign for the Philosophy of Real Mathematics. Having proposed this as a title for my talk, it was sensibly suggested to me that I should moderate its provocative tone, and hence the present version. It is generally an indication of a delusional state to believe without first checking that you are the first to use an expression. The case of ‘real mathematics’ would have proved no exception. In the nineteenth century Kronecker spoke of ‘die wirkliche Mathematik’ to distinguish his algorith- mic style of mathematics from Dedekind’s postulation of infinite collec- tions. But we may also find instances which stand in need of no translation. Listen to G. H. Hardy in A Mathematician’s Apology: It is undeniable that a good deal of elementary mathematics – and I use the word ‘elementary’ in the sense in which professional mathematicians use it, in which it includes, for example, a fair working knowledge of the differential and integral calculus – has considerable practical utility.